Math Core

Multiplying does not always make a number bigger. Half of 12 is 6, and 6 is smaller than 12. In this lesson you will learn to tell whether a product is bigger or smaller than a number without doing the multiplication, just by looking at the other factor.

Resizing a number

Think of multiplying as scaling: stretching or shrinking a number. Watch what happens to 12:

$12 \times \dfrac{1}{2} = 6$ . Shrunk to half its size.
$12 \times 1 = 12$ . Stayed the same.
$12 \times \dfrac{3}{2} = 18$ . Stretched to one and a half times its size.

12 scaled by 1/2, by 1, and by 3/2.

The factor you multiply by decides what happens. The key is to compare that factor to 1.

Scaling rule

When you multiply a number (greater than 0) by a factor:

If the factor is greater than 1, the product is greater than the number.
If the factor is less than 1, the product is less than the number.
If the factor is equal to 1, the product is equal to the number.

Why? Taking $\dfrac{3}{4}$ of something means taking part of it, so you get less than the whole thing. Taking $\dfrac{5}{4}$ of something means the whole thing plus another $\dfrac{1}{4}$ of it, so you get more.

Connection to equivalent fractions

A fraction like $\dfrac{3}{3}$ or $\dfrac{5}{5}$ equals 1. So multiplying by it doesn't change a number's size:

\frac{2}{3} \times \frac{5}{5} = \frac{10}{15}

That is exactly how you make equivalent fractions. $\dfrac{10}{15}$ is the same amount as $\dfrac{2}{3}$ , because you multiplied by 1.

Worked example: Compare without multiplying

Is $\dfrac{4}{5} \times 30$ greater than, less than, or equal to 30?

The factor $\dfrac{4}{5}$ is less than 1, so the product is less than 30. (It is 24, but you did not need to find that.)

Worked example: Which product is greater?

Which is greater, $7 \times \dfrac{2}{3}$ or $7 \times \dfrac{5}{3}$ ?

$\dfrac{2}{3}$ is less than 1, so $7 \times \dfrac{2}{3}$ is less than 7. $\dfrac{5}{3}$ is greater than 1, so $7 \times \dfrac{5}{3}$ is greater than 7. So $7 \times \dfrac{5}{3}$ is greater.

Worked example: Scaling a drawing

A poster is 18 inches tall. A copy is made that is $1\dfrac{1}{3}$ times as tall. Is the copy taller or shorter? How tall is it?

$1\dfrac{1}{3}$ is greater than 1, so the copy is taller. Its height is

1\frac{1}{3} \times 18 = \frac{4}{3} \times 18 = \frac{72}{3} = 24 \text{ inches}

Common mistake

Don't assume "multiply means bigger." That is only true when the factor is greater than 1. $\dfrac{9}{10} \times 50$ is less than 50, even though you multiplied.

Tip

Use scaling to check your answers. If you find $\dfrac{3}{8} \times 16$ and get something bigger than 16, you know you made a mistake.

Practice

Practice 1

Without multiplying, compare $\dfrac{5}{6} \times 42$ to 42.

Practice 2

Without multiplying, compare $1\dfrac{1}{2} \times 8$ to 8.

Practice 3

Without multiplying, compare $9 \times \dfrac{6}{6}$ to 9.

Practice 4

Without multiplying, which product is the greatest?

Practice 5

A drawing of a bridge is 18 centimeters long. It is shrunk to $\dfrac{2}{3}$ of its size. How many centimeters long is the new drawing?

Type a number.

Practice 6

A tree is $1\dfrac{1}{4}$ times as tall as a 12-foot shed. How tall is the tree, in feet?

Type a number.

Practice 7

Leo says, " $\dfrac{3}{4} \times 5\dfrac{1}{2}$ must be greater than $5\dfrac{1}{2}$ , because multiplying makes numbers bigger." Is Leo right?

Practice 8

Put these in order from least to greatest, without multiplying: $P = 20 \times \dfrac{5}{4}$ , $Q = 20 \times \dfrac{3}{5}$ , $R = 20$ .